Phase diagram for the tap energy of the $p$-spin spherical mean field spin glass model

TL;DR

This paper analyzes the phase diagram of the TAP energy in the spherical p-spin spin glass model, revealing distinct high and low temperature phases with different behaviors of TAP solutions and free energy.

Contribution

It provides a detailed solution to the TAP variational principle and characterizes the phase diagram, including the number and nature of TAP maximizers across phases.

Findings

High temperature phase has three subphases with different TAP maximizer behaviors.

Low temperature phase dominated by a subexponential number of near-maximal TAP solutions.

Explicit formulas for the maximum of the variational principle in different phases.

Abstract

We solve the Thouless-Anderson-Palmer (TAP) variational principle associated to the spherical pure $p$-spin mean field spin glass Hamiltonian and present a detailed phase diagram. In the high temperature phase the maximum of variational principle is the annealed free energy of the model. In the low temperature phase the maximum, for which we give a formula, is strictly smaller. The high temperature phase consists of three subphases. (1) In the first phase $m=0$ is the unique relevant TAP maximizer. (2) In the second phase there are exponentially many TAP maximizers, but $m=0$ remains dominant. (3) In the third phase, after the so called dynamic phase transition, $m=0$ is no longer a relevant TAP maximizer, and exponentially many non-zero relevant TAP solutions add up to give the annealed free energy. Finally in the low temperature phase a subexponential number of TAP maximizers of near-maximal TAP energy dominate.